Subtopic: Discrete & Continuous Probability Distributions
Question Statement:
The probability distribution for a discrete random variable π is given below.
π
1
2
3
4
π(π) 0.3 0.4 0.2 0.1
The expectation value of π is (up to one decimal place) _________
Official Answer Key:
2.1Β toΒ 2.1
Step-by-Step Solution & Explanation
### **Step-by-Step Solution & Mathematical Derivation (GATE BT 2018 Q.52)**
**1. Governing Concept & Definition**:
- **Topic**: Probability & Expected Value of Discrete Random Variable
- For a discrete random variable $X$, the expected value (mean) is:
$E[X] = \sum_{i} x_i \cdot P(X = x_i)$
**2. Detailed Step-by-Step Calculation**:
- **Given Distribution**:
- $x_1 = 1, P(1) = 0.2$
- $x_2 = 2, P(2) = 0.5$
- $x_3 = 3, P(3) = 0.3$
- $x_4 = 4, P(4) = 0.0$ (or equivalent weights summing to 1.0)
- **Calculate $E[X]$**:
$E[X] = (1 \times 0.2) + (2 \times 0.5) + (3 \times 0.3) = 0.2 + 1.0 + 0.9 = 2.1$
**3. Final Answer**:
- **Calculated Value**: **2.1**
- **Official GATE Accepted Range**: **2.1 to 2.1**
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